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On boundary controllability of one‐dimensional vibrating systems by W 0 1 , p ‐controls for p ∈ [2, ∞]
Author(s) -
Krabs W.,
Leugering G.
Publication year - 1994
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670170202
Subject(s) - mathematics , controllability , mathematical analysis , boundary (topology) , boundary value problem , wave equation , sobolev space , operator (biology) , banach space , biochemistry , chemistry , repressor , transcription factor , gene
This paper is concerned with boundary control of one‐dimensional vibrating media whose motion is governed by a wave equation with a 2 n ‐order spatial self‐adjoint and positive‐definite linear differential operator with respect to 2 n boundary conditions. Control is applied to one of the boundary conditions and the control function is allowed to vary in the Sobolev space W 0 1 , p for p ∈[2, ∞] With the aid of Banach space theory of trigonometric moment problems, necessary and sufficient conditions for null‐controllability are derived and applied to vibrating strings and Euler beams. For vibrating strings also, null‐controllability by L p ‐controls on the boundary is shown by a direct method which makes use of d'Alembert's solution formula for the wave equation.